On local root numbers and distinction

Abstract

In the context of GLn over a quadratic extension of p-adic fields (p odd) we show that the Rankin-Selberg epsilon factor at 1 2 of a pair of distinguished irreducible representations equals 1. In his dissertation [Ok97], under the supervision of Herve Jacquet, Youngbin Ok indicated an interesting relation between distinction of representations and special values of local Rankin-Selberg gamma factors. In a special case, Ok characterizes distinction in terms of the values of gamma factors. The results in Ok’s dissertation were never published and Jacquet kindly suggested to extend his results. In this work we generalize one direction in Ok’s characterization. We hope to address the other direction in the future. Let E be a local non-archimedean field and ψ a non-trivial character of E. For positive integers r and t and for smooth irreducible representations π of GLr(E) and π ′ of GLt(E) Jacquet, Piatetskii-Shapiro and Shalika attached in [JPSS83] a local Rankin-Selberg Lfactor L(s, π × π′) and -factor (s, π × π′;ψ). We also set (0.1) γ(s, π × π′;ψ) = L(1− s, π × π ′) (s, π × π′;ψ) L(s, π × π′) where π is the contragredient of π. Let E/F be a quadratic extension of non-archimedean local fields. Assume that the characteristic of the residual field of F is odd. A representation (π, V ) of GLr(E) is called GLr(F )-distinguished if there exists a non-zero linear form μ : V → C such that μ(π(h)v) = μ(v), v ∈ V, h ∈ GLr(F ). We may now state the main result of this work (which is [Ana08, Conjecture 5.1]). Theorem 0.1. Let π (resp. π′) be a smooth, irreducible and GLr(F )-distinguished (resp. GLt(F )-distinguished) representation of GLr(E) (resp. GLt(E)). If ψ is a non-trivial character of E with a trivial restriction to F then ( 1 2 , π × π′;ψ) = γ( 2 , π × π′;ψ) = 1. Let us make a few straightforward observations first. The -factor satisfies the identity (0.2) (s, π × π′;ψ) (1− s, π × π′;ψ−1) = 1. Date: November 15, 2009. This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 88/08). Mathematics Subject Classification: primary 11F70; secondary 11F67. 1